Problem: Let $h$ be a twice differentiable function, and let $h(-4)=-3$, $h'(-4)=0$, and $h''(-4)=0$. What occurs in the graph of $h$ at the point $(-4,-3)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-4,-3)$ is a minimum point. (Choice B) B $(-4,-3)$ is a maximum point. (Choice C) C There's not enough information to tell.
Since $h'(-4)=0$, we know that $x=-4$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $h$ at this point according to these three cases: If $h''(-4)>0$, the graph of $h$ has a minimum point at $x=-4$. If $h''(-4)<0$, the graph of $h$ has a maximum point at $x=-4$. If $h''(-4)=0$, the test is inconclusive. [Why is this so?] We are given that $h''(-4)=0$. The test is inconclusive. There's not enough information to tell whether $(-4,-3)$ is a minimum point, a maximum point, or neither.